ELA, Volume 16, pp. 366-379, October 2007, abstract.
Principal Eigenvectors of Irregular Graphs
Sebastian M. Cioaba and David A. Gregory
Let G be a connected graph. This paper studies the extreme entries
of the principal eigenvector x of G, the unique positive unit eigenvector
corresponding to the greatest eigenvalue lambda_1 of the adjacency matrix
of G. If G has maximum degree Delta, the greatest entry x_max of x is
at most 1/sqrt(1+(lambda_1)^2/Delta). This improves a result of Papendieck
and Recht. The least entry x_min of x as well as the principal ratio
x_max/x_min are studied. It is conjectured that for connected graphs of
order n >= 3, the principal ratio is always attained by one of the lollipop
graphs obtained by attaching a path graph to a vertex of a complete graph.